Question

What is integration?

Answer 1

Roughly speaking, integration is the inverse of differentiation, but there are several ways to think about it...

Explanation:

Given a suitably well behaved function `f(x):RR->RR`, and an interval `(a, b)`, the definite integral `int_a^b f(x) dx` is the "area under the curve" between `a` and `b`.

At any particular point `t in RR`, the rate of change of area as you increase `t` is equal to the `f(t)`. That is, the derivative of the integral is equal to the original function.

Integration covers a lot more cases than just Real valued functions of Real numbers. You can integrate over any kind of measurable set - e.g. a plane, a curve, a surface, a volume. The function that you are integrating may have any kind of value that is possible to sum and multiply by a scalar, e.g. Real, Complex, vector.

In such contexts you can think of an integral as a sort of infinite sum of values of a function over infinitesimally small pieces of the set over which you are integrating.

For example, suppose you have a function `f(p)` defined for points on the surface `S` of a sphere, with surface area `A`. Then the average value of `f(p)` over the surface of the sphere is:

`(int_(p in S) f(p) dp) / A`

If we split the surface of the sphere into a large number of little patches `S_i` of areas `A_i`, each containing a representative point `p_i in S_i`, then we could approximate the integral over the surface:

`int_(p in S) f(p) dp ~~ sum_i A_i f(p_i)`